Rank four valinorsmic temperament | Manwe | Varda | Mandos | Ulmo

The valinorsmic rank three temperaments temper out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does.)

Rank four valinorsmic temperament

Comma: 176/175

Map: [<1 0 0 0 -4|, <0 1 0 0 0|, <0 0 1 0 2|, <0 0 0 1 1|]
EDOs: 22, 31, 46, 53, 58, 80, 111

Manwe

Commas: 176/175, 1331/1323

Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
EDOs: 15, 31, 46, 77, 80, 111
Badness: 0.000679

13-limit

Commas: 176/175, 1331/1323, 351/350

Map: [<1 0 0 12 8|, <0 1 0 3 3|, <0 0 1 -6 -4|]
EDOs: 31, 46, 77, 80, 111
Badness: 0.001147

Varda

Commas: 176/175, 896/891

Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1|]
EDOs: 12, 22, 46, 58, 80, 126
Badness: 0.000728

13-limit

Commas: 176/175, 352/351, 364/363

Map: [<2 0 11 0 14 24|, <0 1 -2 0 -4 -7|, <0 0 0 1 1 1|]
EDOs: 22, 46, 58, 80, 138cde, 242cde, 322bcde
Badness: 0.000970

Mandos

Commas: 176/175, 243/242

Map: [<1 1 0 6 2|, <0 2 0 5 5|, <0 0 1 -2 0|]
EDOs: 7, 24, 31, 58, 89
Badness: 0.000751

13-limit

Commas: 144/143, 176/175, 243/242

Map: [<1 1 0 6 2 4|, <0 2 0 5 5 -1|, <0 0 1 -2 0 0|]
EDOs: 7, 24, 31, 58
Badness: 0.000923

Ulmo

Commas: 176/175, 2200/2187

Map: [<1 0 0 1, -3|, <0 1 0 7 7|, <0 0 1 -4 -2|]
EDOs: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd
Badness: 0.00123

13-limit

Commas: 169/168, 176/175, 325/324

Map: [<1 0 0 1, -3 2|, <0 1 0 7 7 4|, <0 0 1 -4 -2 -2|]
EDOs: 7, 39, 46, 53, 80, 133d, 179, 312cdf
Badness: 0.000932